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This paper was commented on through Cosmo Comments. The following comments can also be viewed as annotations on the paper via Hypothesis.


This is an excellent article, rigorous and clearly written, which I would like to recommend to anyone interested in theoretical aspects of gravitational lensing. In my opinion, the highlights are:

• A general method to exactly determine the propagation of scalar, electromagnetic and gravitational waves in space-times which are related to other "known" space-times via the transformation (14).
• An approximate version of it (end of section 3, and nicely illustrated in section 8), which shows in principle how to determine the propagation of waves in any space-time from their behavior in Minkowski.

Hereafter are some more specific comments or questions to the author.

1. Page 3, last sentence of the paragraph after Eq. (7): The author argues that the source term on the right-hand side of (7) may be interpreted as being due to interference between neighboring rays. Could the author further explain that point? Same question for the remark after (108).
2. Equation (14): This transformation is the core of the article. The author nicely explained its geometric meaning in appendix A. However, although I do trust the author on that point, the fact that such a metric transformation is the most general which preserves light rays does not seem to be explicitly proved in the article.
3. End of section 3: Here the author somehow generalizes the previous results to any space-time, arguing that null waves in any space-time can be obtained from their counterpart in Minkowski space-time via diffeomorphisms. Is that equivalent to saying that, in a finite region of space-time, one can pick a coordinate system such that waves propagate in straight lines? If so, how does this method relate to Maartens’ observational coordinates, for instance, or the more recent geodesic light-cone method? In that context, the practical difficulty consists in finding the diffeomorphism leading to the desired metric. Does that also apply here?
4. Section 8. This is really nice!

[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]

Thanks for the comments. The questions were addressed in the revised (and published) version of the paper, but the direct answers are as follows:

The author argues that the source term on the right-hand side of (7) may be interpreted as being due to interference between neighboring rays. Could the author further explain that point? Same question for the remark after (108).

The source term referred to here shows that, e.g., the subleading amplitude is sourced by a coupling (via transverse derivatives) between neighboring leading-order rays, and my interpretation for that was that neighboring rays interfere to determine the first corrections to geometric optics.

Equation (14): This transformation is the core of the article. The author nicely explained its geometric meaning in appendix A. However, although I do trust the author on that point, the fact that such a metric transformation is the most general which preserves light rays does not seem to be explicitly proved in the article.

The equation referred to here describes the metric transformations which preserve optical rays. The starting assumption is that there's some eikonal whose associated wavevector [math]k_a is null with respect to a metric [math]g_{ab}. Changing the metric to [math]\hat{g}_{ab} then results in the same optical rays iff [math]g^{ab} k_b and [math]\hat{g}^{ab} k_b are proportional. That condition also ensures that the rays remain null with respect to the hatted metric.

Now, the most general hatted metric with this property can be found by brute force: a null tetrad can be fixed, the most general real, symmetric rank-2 tensor can be written down in terms of that tetrad, and then the proportionality constraint can be enforced. The result involves 7 free functions. The functions which arise most easily with this method don't have particularly interesting geometrical or physical interpretations. However, an explicit transformation can be found which results in a different set of free functions, each of which does have a clear geometric interpretation in terms of Kerr-Schild, conformal, and related transformations. That final result is what was referred to in the question.

End of section 3: Here the author somehow generalizes the previous results to any space-time, arguing that null waves in any space-time can be obtained from their counterpart in Minkowski space-time via diffeomorphisms. Is that equivalent to saying that, in a finite region of space-time, one can pick a coordinate system such that waves propagate in straight lines? If so, how does this method relate to Maartens’ observational coordinates, for instance, or the more recent geodesic light-cone method? In that context, the practical difficulty consists in finding the diffeomorphism leading to the desired metric. Does that also apply here?

Yes, one consequence is that there are coordinate systems in which at least one system of null rays look like straight lines. Observational coordinates and geodesic light-cone coordinates are essentially versions of this where the chosen null rays are the past-directed null geodesics emanating from an observer's worldline.

As for finding the right diffeo to make this happen, that is usually difficult to do exactly. But there are interesting cases where it's relatively easy to find the diffeo perturbatively.